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Minimax algebra and applications. (English) Zbl 0739.90073
Summary: We consider theories of linear and of polynomial algebra, over two scalar systems, often called max-algebra and min-algebra. Here, max-algebra is the system \(M=\left(\mathbb R\cup\{-\infty\},\oplus,\otimes\right)\) where \(x\oplus y=\max(x,y)\) and \(x\otimes y=x+y\). Min-algebra is the dual system \(M'=\left(\mathbb R\cup\{+\infty\},\oplus',\otimes'\right)\) with \(x\oplus' y=\min(x,y)\) and \(x\otimes'y=x+y\). Towards the end we also consider minimax algebra, the system \(M''=\left(\mathbb R\cup\{-\infty,+\infty\},\, \oplus,\otimes,\oplus',\otimes'\right)\). Application fields discussed include location problems, machine scheduling, cutting and packing problems, discrete-event systems and path-finding problems.

15A80 Max-plus and related algebras
90B80 Discrete location and assignment
90B35 Deterministic scheduling theory in operations research
90C27 Combinatorial optimization
93C83 Control/observation systems involving computers (process control, etc.)
Full Text: DOI
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